Slide 4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4-5 Greatest Common Divisor and Least Common Multiple Definition The greatest common divisor (GCD) of two integers a and b is the greatest integer that divides both a and b. The largest rod that we can use multiples of to build both the 12 and 16 rod is 4. So the GCD(12, 16) = 4. Colored Rods Method Slide 4- 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Intersection on-of-Sets Method Finding the Greatest Common Divisor First, find the sets of all possible divisors of 12 and 16. The set of common positive divisors of 12 and 16 is: Because 4 is the greatest number in this set of common divisors, the GCD of 12 and 16, written GCD(12, 16) = 4. Slide 4- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Prime Factorization Method Finding the Greatest Common Divisor Find the greatest common divisor of 140 and 120. The prime factorizations are: The most factors in common are: GCD(140,120) = 2×2 ×5 = 20 Slide 4- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Euclidean Algorithm Method Finding the Greatest Common Divisor Find the GCD of 1492 and 1008. Slide 4- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Euclidean Algorithm Method Finding the Greatest Common Divisor Find the GCD of 1492 and 1008. GCD(1492, 1008) = GCD(1008, 484) GCD(1008, 484) = (484, 40) GCD(484, 40) = GCD(40, 4) GCD(40, 4) = GCD(4, 0) = 4 Slide 4- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding the Greatest Common Divisor Theorem 4-7
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Slide 4- 1Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4-5 Greatest Common Divisor and Least Common Multiple
DefinitionThe greatest common divisor (GCD) of two integers a and bis the greatest integer that divides both a and b.
The largest rod that we can use multiples of to build both the 12 and 16 rod is 4. So the GCD(12, 16) = 4.
Colored Rods Method
Slide 4- 2Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Intersection on-of-Sets Method
Finding the Greatest Common Divisor
First, find the sets of all possible divisors of 12 and 16.
The set of common positive divisors of 12 and 16 is:
Because 4 is the greatest number in this set of common divisors, the GCD of 12 and 16, written GCD(12, 16) = 4.
Slide 4- 3Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Prime Factorization Method
Finding the Greatest Common Divisor
Find the greatest common divisor of 140 and 120.
The prime factorizations are:
The most factors in common are:
GCD(140,120) = 2×2 ×5 = 20
Slide 4- 4Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Euclidean Algorithm MethodFinding the Greatest Common Divisor
Find the GCD of 1492 and 1008.
Slide 4- 5Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Euclidean Algorithm MethodFinding the Greatest Common Divisor
Find the GCD of 1492 and 1008.
GCD(1492, 1008) = GCD(1008, 484)
GCD(1008, 484) = (484, 40)
GCD(484, 40) = GCD(40, 4)
GCD(40, 4) = GCD(4, 0) = 4
Slide 4- 6Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding the Greatest Common Divisor
Theorem 4-7
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Slide 4- 7Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Definition
Least Common Multiple
Suppose that a and b are positive integers. Then the least common multiple (LCM) of a and b is the least positive integer that is simultaneously a multiple of a and a multiple of b.
Find the Least Common Multiple of 6 and 8.
We are looking for the least number that is a multiple of 6 and a multiple of 8. Stated another way, we are looking for the smallest positive integer divisible by both 6 and 8.
Slide 4- 8Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Colored Rods Method
Least Common Multiple
We build trains of 6 rods and 8 rods until they are the same length.
Least common multiple of 6 and 8 is 24
Slide 4- 9Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Intersection-of-Sets Method
Least Common Multiple
Find the least common multiple of 12 and 16.
First, find the set of all positive multiples of both numbers:
The set of common multiples is:
Since the least of these elements is 48, the least common multiple of 12 and 16 is 48, written LCM(12, 16) = 48.
Slide 4- 10Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Intersection-of-Sets Method
Least Common Multiple
Find the least common multiple of 12 and 16.
First, find the set of all positive multiples of both numbers:
The set of common multiples is:
Since the least of these elements is 48, the least common multiple of 12 and 16 is 48, written LCM(12, 16) = 48.
Slide 4- 11Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Prime Factorization MethodLeast Common Multiple
Find the least common multiple of 12 and 16.
Find the prime factorization of each number:
Take each of the primes that are factors of either number, which in this case would be 2 and 3.
Raise each of these primes to the greatest power of the prime that occurs in either of the prime factorizations. The product is the least common multiple.
Slide 4- 12Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Relationship Between GCD and LCM
Given 12 and 15, first find the prime factorization of both numbers:
Notice that 12 • 16 = GCD(12, 16) • LCM(12, 16)
= 4 • 48
= 192
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Slide 4- 13Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Relationship Between GCD and LCM
For any two natural numbers a and b,
GCD(a,b) • LCM(a,b) = ab.
Theorem 4-8
Suppose we wish to find the least common multiple of 640 and 1296.Applying the Euclidean Algorithm, we can determine that GCD(640, 1296) = 16.
From Theorem 4-8, we know that
16 • LCM(640, 1298) = 640 • 1296
LCM(640, 1296) = (640 • 1296) / 16 = 51,840
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